Financial Engineering
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Stock prices Apple Inc. and Lehman Brothers Inc. in May 2008before the Bankruptcy of Lehman Brothers.
CSE: Financial Engineering Track
Financial Engineering
Financial Engineering combines any fields of mathematics andcomputer science:
Complicated Financial Structures: (CDOs)
Obligor 1 →Obligor 2 →
.
.
.
↓
Obligor m →
PortfolioBond 1Bond 2
.
.
.
↓
Bond n
Periodicpayments−→
←−Sp. payment
(Cash)
SPV
Periodic couponpayments−→
←−Initial cashinvestment
Super Senior TrancheLowest return/Residual loss
Senior Tranche
2nd lowest return/3rd ..% of loss
Mezzanine Tranche
2nd highest return/2nd ..% of loss
Equity TrancheHighest return/1st ..% of loss
Partial (Integro-)Differential Equations: (HeatEquation)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Stochastic Processes: (Brownian Motion, Levyprocesses)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2.5
−2
−1.5
−1
−0.5
0
0.5
1
Time
Pure drift
BM
Compound Poisson
Algorithms: (PSOR)
Choose an initial guess x0 ≥ c.Choose ω ∈ (0, 1] and ε > 0.For k = 0, 1, 2,
For i = 1, . . . , N ,
xk+1i :=
1
Aii
(bi −
i−1∑j=1
Aijxk+1j −
N∑j=i+1
Aijxkj
)xk+1i := max
{ci, x
ki + ω(xk+1
i − xki )}
Next i
If ‖xk+1 − xk‖2 < ε stop elseNext k
CSE: Financial Engineering Track
Financial Engineering
Financial Engineering combines any fields of mathematics andcomputer science:
Complicated Financial Structures: (CDOs)
Obligor 1 →Obligor 2 →
.
.
.
↓
Obligor m →
PortfolioBond 1Bond 2
.
.
.
↓
Bond n
Periodicpayments−→
←−Sp. payment
(Cash)
SPV
Periodic couponpayments−→
←−Initial cashinvestment
Super Senior TrancheLowest return/Residual loss
Senior Tranche
2nd lowest return/3rd ..% of loss
Mezzanine Tranche
2nd highest return/2nd ..% of loss
Equity TrancheHighest return/1st ..% of loss
Partial (Integro-)Differential Equations: (HeatEquation)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Stochastic Processes: (Brownian Motion, Levyprocesses)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2.5
−2
−1.5
−1
−0.5
0
0.5
1
Time
Pure drift
BM
Compound Poisson
Algorithms: (PSOR)
Choose an initial guess x0 ≥ c.Choose ω ∈ (0, 1] and ε > 0.For k = 0, 1, 2,
For i = 1, . . . , N ,
xk+1i :=
1
Aii
(bi −
i−1∑j=1
Aijxk+1j −
N∑j=i+1
Aijxkj
)xk+1i := max
{ci, x
ki + ω(xk+1
i − xki )}
Next i
If ‖xk+1 − xk‖2 < ε stop elseNext k
CSE: Financial Engineering Track
Financial Engineering
Financial Engineering combines any fields of mathematics andcomputer science:
Complicated Financial Structures: (CDOs)
Obligor 1 →Obligor 2 →
.
.
.
↓
Obligor m →
PortfolioBond 1Bond 2
.
.
.
↓
Bond n
Periodicpayments−→
←−Sp. payment
(Cash)
SPV
Periodic couponpayments−→
←−Initial cashinvestment
Super Senior TrancheLowest return/Residual loss
Senior Tranche
2nd lowest return/3rd ..% of loss
Mezzanine Tranche
2nd highest return/2nd ..% of loss
Equity TrancheHighest return/1st ..% of loss
Partial (Integro-)Differential Equations: (HeatEquation)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Stochastic Processes: (Brownian Motion, Levyprocesses)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2.5
−2
−1.5
−1
−0.5
0
0.5
1
Time
Pure drift
BM
Compound Poisson
Algorithms: (PSOR)
Choose an initial guess x0 ≥ c.Choose ω ∈ (0, 1] and ε > 0.For k = 0, 1, 2,
For i = 1, . . . , N ,
xk+1i :=
1
Aii
(bi −
i−1∑j=1
Aijxk+1j −
N∑j=i+1
Aijxkj
)xk+1i := max
{ci, x
ki + ω(xk+1
i − xki )}
Next i
If ‖xk+1 − xk‖2 < ε stop elseNext k
CSE: Financial Engineering Track
Financial Engineering
Financial Engineering combines any fields of mathematics andcomputer science:
Complicated Financial Structures: (CDOs)
Obligor 1 →Obligor 2 →
.
.
.
↓
Obligor m →
PortfolioBond 1Bond 2
.
.
.
↓
Bond n
Periodicpayments−→
←−Sp. payment
(Cash)
SPV
Periodic couponpayments−→
←−Initial cashinvestment
Super Senior TrancheLowest return/Residual loss
Senior Tranche
2nd lowest return/3rd ..% of loss
Mezzanine Tranche
2nd highest return/2nd ..% of loss
Equity TrancheHighest return/1st ..% of loss
Partial (Integro-)Differential Equations: (HeatEquation)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Stochastic Processes: (Brownian Motion, Levyprocesses)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2.5
−2
−1.5
−1
−0.5
0
0.5
1
Time
Pure drift
BM
Compound Poisson
Algorithms: (PSOR)
Choose an initial guess x0 ≥ c.Choose ω ∈ (0, 1] and ε > 0.For k = 0, 1, 2,
For i = 1, . . . , N ,
xk+1i :=
1
Aii
(bi −
i−1∑j=1
Aijxk+1j −
N∑j=i+1
Aijxkj
)xk+1i := max
{ci, x
ki + ω(xk+1
i − xki )}
Next i
If ‖xk+1 − xk‖2 < ε stop elseNext k
CSE: Financial Engineering Track
Different Challenges
I Modelling ChallengesI Modelling uncertain evolution of the market/stocks prices
(Stochastic Processes)
I Default scenarios and BankruptcyI Dependence structure between stock prices
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3Two independent Kou Processes
Time
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4
0.6
0.8
1
1.2
1.4
1.6
1.8Model for Asset Price Processes
Time
X 1t
X 2t
S 1t
S 2t
Path simulation of a two dimensional Kou model.
CSE: Financial Engineering Track
Different Challenges
I Modelling ChallengesI Modelling uncertain evolution of the market/stocks prices
(Stochastic Processes)I Default scenarios and Bankruptcy
I Dependence structure between stock prices0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3Two independent Kou Processes
Time
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4
0.6
0.8
1
1.2
1.4
1.6
1.8Model for Asset Price Processes
Time
X 1t
X 2t
S 1t
S 2t
Path simulation of a two dimensional Kou model.
CSE: Financial Engineering Track
Different Challenges
I Modelling ChallengesI Modelling uncertain evolution of the market/stocks prices
(Stochastic Processes)I Default scenarios and BankruptcyI Dependence structure between stock prices
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3Two independent Kou Processes
Time
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4
0.6
0.8
1
1.2
1.4
1.6
1.8Model for Asset Price Processes
Time
X 1t
X 2t
S 1t
S 2t
Path simulation of a two dimensional Kou model.
CSE: Financial Engineering Track
Different Challenges
I Modelling ChallengesI Modelling uncertain evolution of the market/stocks prices
(Stochastic Processes)I Default scenarios and BankruptcyI Dependence structure between stock prices
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3Two independent Kou Processes
Time
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4
0.6
0.8
1
1.2
1.4
1.6
1.8Model for Asset Price Processes
Time
X 1t
X 2t
S 1t
S 2t
Path simulation of a two dimensional Kou model.
CSE: Financial Engineering Track
Different Challenges
I Mathematical ChallengesI Casting the stochastic models into a mathematical framework
(Stochastic Calculus)
I Relation between stochastic stock-price processes and value offinancial contracts
I To determine the value of financial contracts non-standardmethods are necessary (Monte Carlo, Finite Elements, FourierMethods)
00.511.522.533.544.55
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Spot s1
Spot s2
Op
tio
n p
rice
Value of an American Basket option in the 2-dimensional CGMYLevy Model.
CSE: Financial Engineering Track
Different Challenges
I Mathematical ChallengesI Casting the stochastic models into a mathematical framework
(Stochastic Calculus)I Relation between stochastic stock-price processes and value of
financial contracts
I To determine the value of financial contracts non-standardmethods are necessary (Monte Carlo, Finite Elements, FourierMethods)
00.511.522.533.544.55
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Spot s1
Spot s2
Op
tio
n p
rice
Value of an American Basket option in the 2-dimensional CGMYLevy Model.
CSE: Financial Engineering Track
Different Challenges
I Mathematical ChallengesI Casting the stochastic models into a mathematical framework
(Stochastic Calculus)I Relation between stochastic stock-price processes and value of
financial contractsI To determine the value of financial contracts non-standard
methods are necessary (Monte Carlo, Finite Elements, FourierMethods)
00.511.522.533.544.55
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Spot s1
Spot s2
Op
tio
n p
rice
Value of an American Basket option in the 2-dimensional CGMYLevy Model.
CSE: Financial Engineering Track
Different Challenges
I Mathematical ChallengesI Casting the stochastic models into a mathematical framework
(Stochastic Calculus)I Relation between stochastic stock-price processes and value of
financial contractsI To determine the value of financial contracts non-standard
methods are necessary (Monte Carlo, Finite Elements, FourierMethods)
00.511.522.533.544.55
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Spot s1
Spot s2
Op
tio
n p
rice
Value of an American Basket option in the 2-dimensional CGMYLevy Model.
CSE: Financial Engineering Track
Different Challenges
I Computational ChallengesI Computation in high dimension (up to 500 dim.) can be tricky
I Fast algorithms are crucial in financial markets (advancedcomputer languages are key)
00.5
11.5
22.5
3
0
0.5
1
1.5
2
2.5
3
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
Spot s1
Delta of S1
Spot s2
00.5
11.5
22.5
3
0
0.5
1
1.5
2
2.5
3
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
Spot s1
Delta of S2
Spot s2
Sensitivity (derivative wrt s1 and s2) of an American Basket optionin the 2-dimensional CGMY Levy Model.
CSE: Financial Engineering Track
Different Challenges
I Computational ChallengesI Computation in high dimension (up to 500 dim.) can be trickyI Fast algorithms are crucial in financial markets (advanced
computer languages are key)
00.5
11.5
22.5
3
0
0.5
1
1.5
2
2.5
3
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
Spot s1
Delta of S1
Spot s2
00.5
11.5
22.5
3
0
0.5
1
1.5
2
2.5
3
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
Spot s1
Delta of S2
Spot s2
Sensitivity (derivative wrt s1 and s2) of an American Basket optionin the 2-dimensional CGMY Levy Model.
CSE: Financial Engineering Track
Different Challenges
I Computational ChallengesI Computation in high dimension (up to 500 dim.) can be trickyI Fast algorithms are crucial in financial markets (advanced
computer languages are key)
00.5
11.5
22.5
3
0
0.5
1
1.5
2
2.5
3
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
Spot s1
Delta of S1
Spot s2
00.5
11.5
22.5
3
0
0.5
1
1.5
2
2.5
3
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
Spot s1
Delta of S2
Spot s2
Sensitivity (derivative wrt s1 and s2) of an American Basket optionin the 2-dimensional CGMY Levy Model.
CSE: Financial Engineering Track
Prospects
Jobs/Career
I Academics (Universities and research institutes, PhD,Post-doc, etc.)
I Banking (Private banking, asset and risk management,financial product development, etc.)
I Investment services (Hedge funds, pension funds, privateequity, etc.)
I Insurance (life-insurance, non-life insurance, Re-insurance,etc.)
I Other financial service industries (high frequency trading, etc)
I Consulting (Model validation, Regulation, etc.)
CSE: Financial Engineering Track
Prospects
Jobs/Career
I Academics (Universities and research institutes, PhD,Post-doc, etc.)
I Banking (Private banking, asset and risk management,financial product development, etc.)
I Investment services (Hedge funds, pension funds, privateequity, etc.)
I Insurance (life-insurance, non-life insurance, Re-insurance,etc.)
I Other financial service industries (high frequency trading, etc)
I Consulting (Model validation, Regulation, etc.)
CSE: Financial Engineering Track
Prospects
Jobs/Career
I Academics (Universities and research institutes, PhD,Post-doc, etc.)
I Banking (Private banking, asset and risk management,financial product development, etc.)
I Investment services (Hedge funds, pension funds, privateequity, etc.)
I Insurance (life-insurance, non-life insurance, Re-insurance,etc.)
I Other financial service industries (high frequency trading, etc)
I Consulting (Model validation, Regulation, etc.)
CSE: Financial Engineering Track
Prospects
Jobs/Career
I Academics (Universities and research institutes, PhD,Post-doc, etc.)
I Banking (Private banking, asset and risk management,financial product development, etc.)
I Investment services (Hedge funds, pension funds, privateequity, etc.)
I Insurance (life-insurance, non-life insurance, Re-insurance,etc.)
I Other financial service industries (high frequency trading, etc)
I Consulting (Model validation, Regulation, etc.)
CSE: Financial Engineering Track
Prospects
Jobs/Career
I Academics (Universities and research institutes, PhD,Post-doc, etc.)
I Banking (Private banking, asset and risk management,financial product development, etc.)
I Investment services (Hedge funds, pension funds, privateequity, etc.)
I Insurance (life-insurance, non-life insurance, Re-insurance,etc.)
I Other financial service industries (high frequency trading, etc)
I Consulting (Model validation, Regulation, etc.)
CSE: Financial Engineering Track
Prospects
Jobs/Career
I Academics (Universities and research institutes, PhD,Post-doc, etc.)
I Banking (Private banking, asset and risk management,financial product development, etc.)
I Investment services (Hedge funds, pension funds, privateequity, etc.)
I Insurance (life-insurance, non-life insurance, Re-insurance,etc.)
I Other financial service industries (high frequency trading, etc)
I Consulting (Model validation, Regulation, etc.)
CSE: Financial Engineering Track